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Positive steady states of a ratio-dependent predator-prey system with cross-diffusion

Xiaoling Li Guangping Hu Xianpei Li Zhaosheng Feng

*Corresponding author: Zhaosheng Feng zhaosheng.feng@utrgv.edu

MBE2019,6,6753doi:10.3934/mbe.2019337

In this paper, we study a ratio-dependent predator-prey system with diffusion and cross-diffusion under the homogeneous Neumann boundary condition. By applying the maximum principle and Harnack’s inequality, we present a priori estimates of the positive steady state of the system. The existence and non-existence of non-constant positive steady states are established. Our findings show that under certain hypotheses, non-constant positive steady states can exist due to the emergence of cross-diffusion, which reveals that cross-diffusion can induce stationary patterns but the random diffusion fails.

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